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Level 4 7. The sine of angle theta is 0.3 . What is cos(theta) ? Explain how you know.

Level 44\newline77. The sine of angle θ \theta is 00.33 .\newlineWhat is cos⁡(θ) \cos (\theta) ? Explain how you know.

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Q. Level 44\newline77. The sine of angle θ \theta is 00.33 .\newlineWhat is cos⁡(θ) \cos (\theta) ? Explain how you know.
  1. Pythagorean Identity: We know that for any angle θ\theta in a right-angled triangle, the sine and cosine are related by the Pythagorean identity: sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1. Given that sin⁡(θ)=0.3\sin(\theta) = 0.3, we can use this identity to find cos⁡(θ)\cos(\theta).
  2. Square Sine: First, we square the sine of theta to get sin⁡2(θ)\sin^2(\theta): (0.3)2=0.09(0.3)^2 = 0.09.
  3. Substitute Identity: Next, we substitute sin⁡2(θ)\sin^2(\theta) into the Pythagorean identity: 0.09+cos⁡2(θ)=10.09 + \cos^2(\theta) = 1.
  4. Solve for Cosine: We then rearrange the equation to solve for cos⁡2(θ)\cos^2(\theta): cos⁡2(θ)=1−0.09=0.91\cos^2(\theta) = 1 - 0.09 = 0.91.
  5. Consider Both Cases: To find cos⁡(θ)\cos(\theta), we take the square root of cos⁡2(θ)\cos^2(\theta). Since cosine can be positive or negative depending on the quadrant of θ\theta, we consider both cases. However, without additional information about the angle θ\theta, we typically take the positive square root for the principal value: cos⁡(θ)=0.91\cos(\theta) = \sqrt{0.91}.
  6. Calculate Cosine: Calculating the square root of 0.910.91 gives us cos⁡(θ)≈0.954\cos(\theta) \approx 0.954.

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